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The Sum Number of a Disjoint Union of Graphs

The Sum Number of a Disjoint Union of Graphs. Mirka Miller & Joe Ryan The University of Newcastle, Australia W. F. Smyth McMaster University, Hamilton, Canada Curtin University, Perth, Australia. Sum Labelling. L : V(G)  ℕ. For u, v  V, (u, v)  E(G) if and only if

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The Sum Number of a Disjoint Union of Graphs

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  1. The Sum Number of a Disjoint Union of Graphs Mirka Miller & Joe Ryan The University of Newcastle, Australia W. F. Smyth McMaster University, Hamilton, Canada Curtin University, Perth, Australia

  2. Sum Labelling L : V(G)  ℕ. For u, v  V, (u, v)  E(G) if and only if  w  V such that L(w) = L(u) + L(v).

  3. Sum Graphs • All sum graphs are disconnected. • Any graph can be made to support a sum labelling by adding sufficient isolated vertices called isolates. • The smallest number of isolates required is called the sum number of the graph (σ(G)). • Sum graphs with this fewest number of isolates are called optimal.

  4. ExampleA Non Optimal Labelling 27 4 9 81 12 3 36 1 84

  5. ExampleA Optimal Sum Labelling 7 14 4 11 3 1

  6. Potential Perils in Sum Labelling 3 1 2 6 4 9 5

  7. Disjoint Union of Graphs(an example) 14 3 5 18 21 29 22 26 18 4 21 9 30 34 38 7 13 11 17

  8. Disjoint Union of Graphs(an example) 98 114 35 126 798 1102 154 182 684 152 147 63 210 238 266 91 418 266 119

  9. An Upper Bound • σ(G1G2)  σ(G1) + σ(G2) – 1 • Inequality is tight for unit graphs • The technique may be applied repeatedly for a disjoint union of many graphs.

  10. Three Unit Graphs: An Example 10 5 2 1 1 1 4 2 3 4 3 3 5 4 7 5 6 14

  11. Three Unit Graphs: An Example 10 5 2 14 1 1 4 2 3 4 3 42 70 56 7 5 84

  12. Three Unit Graphs: An Example 10 5 168 14 84 1 4 2 3 336 252 42 70 56 7 420 A disjoint union of three graphs with sum number 1

  13. A Disjoint Union of p Graphs(main result) Provided that we can always find a label in one graph that is co-prime to the largest label in one of the others. Easy if 1 is a label in any of the graphs.

  14. Can we always apply the co-prime condition? • Yes if 1 is a label of any of the graphs. • No sum graph has yet been found that cannot bear a sum labelling containing 1. • But… “absence of evidence is not evidence of absence” Rumsfeld • Exclusive sum graphs may always be labelled with a labelling scheme containing 1.

  15. Exclusive Sum Graphs • If L is an exclusive sum labelling for a graph G, so is k1L+k2 where k1, k2are integers such that min(k1L+k2)  1. Miller, Ryan, Slamin, Sugeng, Tuga (2003) Provided at least one of the graphs is an exclusive graph

  16. Open Questions • Can we always find a sum labelling containing the label 1? • What is the sum number of a disjoint union of graphs for various families of graphs? • What is the exclusive sum number of a disjoint union of graphs for various families of graphs?

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